Can fully connected layers be used for feature detection?
First of all, (FC) Neural Networks (NN) are universal function approximators. This means, that, in theory, there must exist some NN of appropriate size that is capable of doing what a CNN can do as well. The only difference is that the standard NN would have to be considerably larger than a corresponding CNN with the same capabilities. Each input pixel would probably have to map to a few dozens to hundreds of input nodes that are trained each on detecting some part of a distinct feature in their respective input image location. Consecutive, hidden layers than have to combine the parts extracted by the input layer for any given location and finally act as the true feature detectors, detecting features as compositions of values in the first layer.
So, for each position in the input image separately, all the filters have to be learned separately. This absence of weight sharing (since filters are not shared across input locations, but have to be trained for each separate input location) makes using a standard NN for feature extraction so inefficient.
However, in theory, also a standard NN could be trained on solving an image classification task, thereby learning to extract features. Actually, I would not be surprised if someone had tested that already on the MNIST dataset.